Here’s the sentence:
“I’m one of those lucky people whose job is an extension of his hobby.”
What’s wrong with it?
Nothing, at least upon first reading.
But as my colleague and brilliant wordsmith, Paul Stregevsky, patiently explained to me, the sentence should be,
“I’m one of those lucky people whose jobs are extensions of their hobbies.”
Huh? (If you have that same reaction, rest assured, you’re in good company.)
Take it away, Paul:
Consider these two sentences, if they were uttered by Melania Trump:
1. I’m one of those women who sometimes wants to murder my husband. (Not grammatical.)
2. I’m one of those women who sometimes want to murder their husbands. (Grammatical.)
3. I’m one of those women who sometimes wants to murder her husband. (Not grammatical.)
4. I’m one of those women who sometimes want to murder my husband. (Grammatical.)
Mind you, sentence 1 is not grammatical. But let’s imagine it were. Do you understand why sentence 1 means something radically different from the meaning she intended?
Sentence 1, if it were grammatical, would have to mean, “Like many women, I sometimes want to murder Donald Trump.” And that's not what you intended, is it?
Sentence 2 means, “Many women sometimes want to murder their own husbands. I, do, too.”
Sentence 3 is a blend of the two: It’s faithful to the grammatical number of sentence 1 and the grammatical person intended by sentence 2. In effect, it splits the difference. But it, too, would be grammatically wrong.
Let’s try it again, with a more perverted scenario:
Sentence 1: I’m one of those dads who sometimes imagines banging my daughter.
Sentence 2: I’m one of those dads who sometimes imagines banging their daughters.
In sentence 1, are you trying to say that many dads want to bang your daughter? Because that’s what it must mean.
Here’s another example:
“I’m a change agent who rolls up my sleeves to get things done.”
Grammatically, the sentence should be,
“I’m a change agent who rolls up his sleeves to get things done.”
Again, here’s Paul:
Well, what a coincidence: I, too, am a change agent who rolls up Jonathan Rick’s sleeves to get things done.
At least, that’s what your wording suggests is possible.
The singular vs. plural choice is merely an artifact of the first-person vs. third-person choice. Once we establish that the grammatical number is driven by whatever agent is doing the rolling up (people), we must follow through and complete the sentence in the same grammatical person (third/he).
In other words: The pronoun must be third-person (“her sleeves”), not first (“my sleeves”).